Embedded Stochastic Runge‐Kutta Methods

We present some new embedded explicit stochastic Runge‐Kutta methods for the approximation of Stratonovich stochastic differential equations in the weak sense with different orders of convergence. The presented methods yield an estimate of the local error which can be used for a step size control algorithm.     

Implicit Stochastic Runge–Kutta Methods for Stochastic Differential Equations

In this paper we construct implicit stochastic Runge–Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Runge‐Kutta methods for monotone differential and stochastic equations

Runge‐Kutta methods which preserve monotonicity for deterministic ordinary differential equations also preserve montonicity for random differential equations albeit with reduced order. However, the only one‐step numerical methods which preserve the montone structure of a monotone stochastic differential equation are the strong Taylor schemes of strong order 0:5 and 1:0.     

Multivalued Stochastic Differential Equations with Non-Lipschitz Coefficients

The existence and uniqueness of solutions to the multivalued stochastic differential equations with non-Lipschitz coefficients are proved, and bicontinuous modifications of the solutions are obtained.     

Support Theorem for Stochastic Differential Equations with Sobolev Coefficients

In this paper we prove a support theorem for stochastic differential equations with Sobolev coefficients in the framework of DiPerna-Lions theory.

     

The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations

In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.     

Fitted modifications of classical Runge‐Kutta pairs of orders 5(4)

Runge‐Kutta pairs sharing orders 5 and 4 are among the most celebrated methods for solving initial value problems. Here, after considering the phase lag as a function of frequency v, we derive a modification of a particular pair. Namely, we present a zero dissipative pair with vanished phase error and its first derivative. After extended numerical tests in various oscillatory problems, it seems that this modified pair outperforms existing methods in the literature.     

非Lipschitz条件下的倒向重随机微分方程

该文研究了非Lipschitz条件下的倒向重随机微分方程, 给出了此类方程解的存在唯一性 定理, 推广Pardoux和Peng 1994年的结论; 同时也得到了此类方程在非Lipschitz条件下的比较定理, 推广了Shi,Gu和Liu 2005年的结果. 从而推广倒向重随机微分方程在随机控制和随机偏微分方程在 粘性解方面的应用.     

Numerical smoothing of Runge–Kutta schemes

First we give an intuitive explanation of the general idea of Sun (2005) [1]: consistency and numerical smoothing implies convergence and, in addition, enables error estimates. Then, we briefly discuss some of the advantages of numerical smoothing over numerical stability in error analysis. The main aim of this paper is to introduce a smoothing function and use it to investigate the smoothing properties of some familiar schemes.     

Runge–Kutta Methods for Itô Stochastic Differential Equations with Scalar Noise

A general class of stochastic Runge–Kutta methods for Itô stochastic differential equation systems w.r.t. a one-dimensional Wiener process is introduced. The colored rooted tree analysis is applied to derive conditions for the coefficients of the stochastic Runge–Kutta method assuring convergence in the weak sense with a prescribed order. Some coefficients for new stochastic Runge–Kutta schemes of order two are calculated explicitly and a simulation study reveals their good performance.     

Low-storage Runge–Kutta methods for stochastic differential equations

Runge–Kutta methods that require only two memory locations per variable and have strong local order γ=1.5 for non-commutative systems of stochastic differential equations driven by one Wiener process are devised in this paper. A first step in the derivation is to extend existing deterministic methods to the commutative stochastic case, for which higher accuracy is also obtained. Numerical results are presented to validate the approach.     

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

An Approximation Scheme for Reflected Stochastic Differential Equations with Non-Lipschitzian Coefficients

Journal of Theoretical Probability - In this paper, we study a numerical approximation scheme for reflected stochastic differential equations (SDEs) with non-Lipschitzian coefficients in a bounded...     

Averaging of Backward Stochastic Differential Equations and Homogenization of Partial Differential Equations with Periodic Coefficients

Abstract

We study the limit of the solutions of systems of semi-linear partial differential equations (PDEs) of second order of parabolic type, with rapidly oscillating periodic coefficients, a singular drift, and singular coefficients of the zero and second order terms. Our basic tool is the approach given by Pardoux [14 Pardoux , E. 1999 . Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach . J. Funct. Anal. 167 : 498 – 520 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]]. In particular, we use the weak convergence of an associated backward stochastic differential equation (BSDE).     

Runge‐Kutta Type Methods for Ill‐Posed Problems and the Retrieval of Aerosol Size Distributions

The present paper deals with new explicit and implicit Runge‐Kutta type iteration methods as a regularization tool for ill‐posed problems. These methods are a generalization of the well‐known Landweber method. An application to Atmospheric Physics in determining the aerosol size distribution is given.     

带奇异系数的有限维与无穷维随机微分方程

本文对具非Lipschitz系数的随机微分方程给出解的存在唯一性与非爆炸性的新判别条件,少许改进了文\cite{4}的有关结果. 通过控制交互作用, 该结果还被推广到无穷维情形.